I initially posted a question and thought that was the only one I was stuck on, but there's this other question too.
I think I kinda solved it, but I'm getting that it isn't divisible by 9. I'd like a confirmation please!
The question is as follows:
Use mathematical induction to prove that $$5^{2n+1} - 21n + 31$$ is divisible by $9$ for $\forall n \geq 1$.
On
For $n = 1$,
$5^{2n+1} - 21n + 31 = 135$, which is divisible by 9.
Assume the claim hold for $n = k$.
When $n = k + 1$,
$5^{2(k+1)+1}−21(k+1)+31 \pmod9$
$\equiv 25(5^{2k+1})−21k + 10 \pmod9$
$\equiv 25(21k - 31) - 21k + 10 \pmod9$
$\equiv 9(56k) -31*25 + 10 \pmod9$
$\equiv 0 \pmod9$
Therefore the claim is true for all integer n $\ge 1$
$5^{2n+1}-21n+31$ is divisible by $9$.
Here is a hint for the induction step:
$5^{2(n+1)+1}-21(n+1)+31=25(5^{2n+1}-21n+31)+504n-765$